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Theorem xpdisj1 4817
Description: Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
Assertion
Ref Expression
xpdisj1  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  C )  i^i  ( B  X.  D ) )  =  (/) )

Proof of Theorem xpdisj1
StepHypRef Expression
1 inxp 4536 . 2  |-  ( ( A  X.  C )  i^i  ( B  X.  D ) )  =  ( ( A  i^i  B )  X.  ( C  i^i  D ) )
2 xpeq1 4423 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  B )  X.  ( C  i^i  D ) )  =  (
(/)  X.  ( C  i^i  D ) ) )
3 0xp 4484 . . 3  |-  ( (/)  X.  ( C  i^i  D
) )  =  (/)
42, 3syl6eq 2133 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  B )  X.  ( C  i^i  D ) )  =  (/) )
51, 4syl5eq 2129 1  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  C )  i^i  ( B  X.  D ) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287    i^i cin 2987   (/)c0 3275    X. cxp 4407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3930  ax-pow 3982  ax-pr 4008
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3416  df-sn 3436  df-pr 3437  df-op 3439  df-opab 3874  df-xp 4415  df-rel 4416
This theorem is referenced by:  djudisj  4820  xpfi  6583  djuinr  6691  djuin  6692
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